- Let g≡and let f=. Find g ∘ f. Include the domain of g ∘ f.
- Give the domains of the following functions.
- f=
- f=
- f=
- f=
- f=

- f
- Let f : ℝ → ℝ be defined by f≡ t
^{3}+ 1. Is f one to one? Can you find a formula for f^{−1}? - Suppose a
_{1}= 1,a_{2}= 3, and a_{3}= −1. Suppose also that for n ≥ 4 it is known that a_{n}= a_{n−1}+ 2a_{n−2}+ 3a_{n−3}. Find a_{7}. Are you able to guess a formula for the k^{th}term of this sequence? - Let f : → ℝ be defined by f≡. Find f
^{−1}if possible. - A function f : ℝ → ℝ is a strictly increasing function if whenever x < y, it follows that
f< f. If f is a strictly increasing function, does f
^{−1}always exist? Explain your answer. - Let fbe defined by
Find f

^{−1}if possible. - Suppose f : D→ Ris one to one, R⊆ D, and g : D→ Ris one to one. Does it follow that g ∘ f is one to one?
- If f : ℝ → ℝ and g : ℝ → ℝ are two one to one functions, which of the following are
necessarily one to one on their domains? Explain why or why not by giving a proof or
an example.
- f + g
- fg
- f
^{3} - f∕g

- Draw the graph of the function f= x
^{3}+ 1. - Draw the graph of the function f= x
^{2}+ 2x + 2. - Draw the graph of the function f=.
- Suppose a
_{n}=and let n_{k}= 2^{k}. Find b_{ k}where b_{k}= a_{nk}. - If X
_{i}are sets and for some j, X_{j}= ∅, the empty set. Verify carefully that ∏_{i=1}^{n}X_{i}= ∅. - Suppose f+ f= 7 x and f is a function defined on ℝ∖, the nonzero real numbers. Find all values of x where f= 1 if there are any. Does there exist any such function?
- Does there exist a function f, satisfying f−f= 3 x which has both x andin the domain of f?
- In the situation of the Fibonacci sequence show that the formula for the n
^{th}term can be found and is given byHint: You might be able to do this by induction but a better way would be to look for a solution to the recurrence relation, a

_{n+2}≡ a_{n}+ a_{n+1}of the form r^{n}. You will be able to show that there are two values of r which work, one of which is r =. Next you can observe that if r_{1}^{n}and r_{ 2}^{n}both satisfy the recurrence relation then so does cr_{1}^{n}+ dr_{2}^{n}for any choice of constants c,d. Then you try to pick c and d such that the conditions, a_{1}= 1 and a_{2}= 1 both hold. - In an ordinary annuity, you make constant payments, P at the beginning of each
payment period. These accrue interest at the rate of r per payment period. This means
at the start of the first payment period, there is the payment P ≡ A
_{1}. Then this produces an amount rP in interest so at the beginning of the second payment period, you would have rP + P + P ≡ A_{2}. Thus A_{2}= A_{1}+ P. Then at the beginning of the third payment period you would have A_{2}+ P ≡ A_{3}. Continuing in this way, you see that the amount in at the beginning of the n^{th}payment period would be A_{n}given by A_{n}= A_{n−1}+ P and A_{1}= P. Thus A is a function defined on the positive integers given recursively as just described and A_{n}is the amount at the beginning of the n^{th}payment period. Now if you wanted to find out A_{n}for large n, how would you do it? One way would be to use the recurrance relation n times. A better way would be to find a formula for A_{n}. Look for one in the form A_{n}= Cz^{n}+ s where C,z and s are to be determined. Show that C =,z =, and s = −. - A well known puzzle consists of three pegs and several disks each of a different diameter,
each having a hole in the center which allows it to be slid down each of the pegs. These
disks are piled one on top of the other on one of the pegs, in order of decreasing
diameter, the larger disks always being below the smaller disks. The problem is to move
the whole pile of disks to another peg such that you never place a disk on a smaller disk.
If you have n disks, how many moves will it take? Of course this depends on n. If
n = 1, you can do it in one move. If n = 2, you would need 3. Let A
_{n}be the number required for n disks. Then in solving the puzzle, you must first obtain the top n − 1 disks arranged in order on another peg before you can move the bottom disk of the original pile. This takes A_{n−1}moves. Explain why A_{n}= 2A_{n−1}+ 1,A_{1}= 1 and give a formula for A_{n}. Look for one in the form A_{n}= Cr^{n}+ s. This puzzle is called the Tower of Hanoi. When you have found a formula for A_{n}, explain why it is not possible to do this puzzle if n is very large.

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